Question

$\sqrt{8 + \sqrt{28} } - \sqrt{8 - \sqrt{28} } \div \sqrt{8 + \sqrt{28} } + \sqrt{8 - \sqrt{28} }$
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Answer 1

Answer:

GIVEN THAT:-

$\frac{(8 + \sqrt{28} ) - (8 - \sqrt{28} )}{(8 + \sqrt{28} ) + (8 + \sqrt{28} )}$

RATIONALIZATION -

$= \frac{(8 + \sqrt{28}) - (8 - \sqrt{28}) }{(8 + \sqrt{28} ) + (8 - \sqrt{28}) } \times \frac{(8 + \sqrt{28}) - (8 + \sqrt{28}) }{(8 + \sqrt{28}) - (8 - \sqrt{28} ) } \\ \\ = \frac{ {((8 + \sqrt{28} ) - (8 - \sqrt{28} ))}^{2} }{ {(8 + \sqrt{28} )}^{2} - {(8 - \sqrt{28} )}^{2} } \\ \\ = \frac{ {(8 + \sqrt{28} ) }^{2} + {(8 - \sqrt{28}) }^{2} - 2(8 + \sqrt{28} )(8 - \sqrt{28} )}{(8 + \sqrt{28} + 8 - \sqrt{28} )(8 + \sqrt{28} - 8 + \sqrt{28} )} \\ \\ = \frac{64 + 28 + 16 \sqrt{28} + 64 + 28 - 16 \sqrt{28} - 2( {(8)}^{2} - 28)}{(16)(2 \sqrt{28} )} \\ \\ = \frac{128 + 56 - 2(64 - 28)}{32 \sqrt{28} } \\ \\ = \frac{184 - 2 \times 36}{32 \sqrt{28} } \\ \\ = \frac{184 - 72}{32 \sqrt{28} } \\ \\ = \frac{112}{32 \sqrt{28} } \\ \\ = \frac{7}{4 \sqrt{7} } \\ \\ = \frac{ \sqrt{7} }{4}$

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